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Modulation For the QR code symbols to be read correctly and quickly, it is important to randomize the two-dimensional recording pattern of the code symbol and also to remove, as much as possible, the speci ed binary pattern of the square-shaped marks from the recording data. To satisfy these, the recording data is modulo-2 added by some speci c patterns, that is, modulation is performed. (For more details, the reader should refer to [JAPA02].) 13.4.2 Two-dimensional Unidirectional Clustered Error Correcting Codes
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Two-dimensional unidirectional clustered error correcting codes are suitable for highdensity two-dimensional matrix symbols [KANE03]. The codes are capable of correcting unidirectional errors con ned to a rectangle having lm rows and ln columns. Because these symbols are usually printed on surfaces such as paper, plastic, and metal, they can sometimes be damaged by blots or scratches, as was mentioned before. Further these errors are unidirectional because blotted white cells change into black cells while black cells remain unchanged, and scratched black cells change into white cells while white cells remain unchanged. These types of errors can be effectively corrected by twodimensional unidirectional clustered error correcting codes. Ef cient one-dimensional unidirectional burst error correcting codes are presented in [PARK90] and [SAIT90b]. Also one-dimensional unidirectional byte error correcting codes have been constructed in [SAOW00]. Two-dimensional clustered error correcting codes, which are capable of correcting any errors con ned in a rectangle with lm rows and ln columns, are presented in [BREI98]. No ef cient two-dimensional unidirectional clustered error correcting codes, however, have been presented. This subsection covers this new class of array codes and shows them to be capable of correcting unidirectional lm ln -clustered errors. 1. One-dimensional Unidirectional Burst Error Correcting Codes The existing one-dimensional unidirectional r-bit burst error correcting codes [PARK90] are abbreviated as 1D-Ur BEC codes. These codes can be used to construct the two-dimensional unidirectional lm ln -clustered error correcting codes, abbreviated as 2D-Ulm ln EC codes. Let Di di;r 1 di;r 2 di;0 be a binary information block having length r bits. The information part of the 1D-Ur BEC code having length k r bits is expressed as Dk 1 Dk 2 D0 . The parity check PC1D and arithmetic residue check ARC1D for Dk 1 Dk 2 D0 are performed as follows: 1 0 1 0 1 1 0 dk 1;r 1 dk 2;r 1 d0;r 1 pr 1 B pr 2 C B dk 1;r 2 C B dk 2;r 2 C B d0;r 2 C B C B C B C B C B C B C B C B C B . C B . . C B C B . C; . . @ . A @ A @ A @ . A . . . . dk 1;0 dk 2;0 d0;0 p0 ! k 1 X i 1 w Di mod 2kr 1 ; 0
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i 0 T where PC1D is a transpose of PC1D , denotes addition over GF 2 , and w Di is the Hamming weight of Di . Let C be a balanced code [PARK90] having jCj ! 2kr 1
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